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This phenomenon is specific to the fields $\mathbb F_q$ (in this case you need the Artin motives as well) and $\bar{\mathbb F}_q$. Roughly, the outline is as follows.

If the Tate conjecture is true and the Frobenius action is semisimple, then the homological standard conjecture is true. Then Jannsen's Motives, numerical equivalence, and semi-simplicity implies that the category of motives with homological equivalence is semisimple. Moreover, the Tate conjecture gives a description of the simple objects in terms of the characteristic polynomial of Frobenius. Then Honda–Tate theory constructs all such characteristic polynomials inside the cohomology of an abelian variety. These are always dominated by Jacobians, so all desired Frobenius eigenvalues occur in the $H^1$ of a curve.

A great general reference for motives is the AMS Proceedings of Symposia in Pure Mathematics Vol. 55.1: Motives. See here for the AMS ebook collection version (you might be able to access this from within your institution). See in particular Proposition 2.6 and Remark 2.7 of Milne's chapter Motives over finite fields.

Remark. As I understand it, the same is not expected to hold over basically any other type of field. For example, it should be "easy" to write down a Hodge structure that is not in the full abelian tensor subcategory generated by the Hodge structures of weight $1$.

This phenomenon is specific to the fields $\mathbb F_q$ (in this case you need the Artin motives as well) and $\bar{\mathbb F}_q$. Roughly, the outline is as follows.

If the Tate conjecture is true and the Frobenius action is semisimple, then the homological standard conjecture is true. Then Jannsen's Motives, numerical equivalence, and semi-simplicity implies that the category of motives with homological equivalence is semisimple. Moreover, the Tate conjecture gives a description of the simple objects in terms of the characteristic polynomial of Frobenius. Then Honda–Tate theory constructs all such characteristic polynomials inside the cohomology of an abelian variety. These are always dominated by Jacobians, so all desired Frobenius eigenvalues occur in the $H^1$ of a curve.

A great general reference for motives is the AMS Proceedings of Symposia in Pure Mathematics Vol. 55.1: Motives. See here for the AMS ebook collection version (you might be able to access this from within your institution). See in particular Proposition 2.6 and Remark 2.7 of Milne's chapter Motives over finite fields (preprint here).

Remark. As I understand it, the same is not expected to hold over basically any other type of field. For example, it should be "easy" to write down a Hodge structure that is not in the full abelian tensor subcategory generated by the Hodge structures of weight $1$.

This phenomenon is specific to the fields $\mathbb F_q$ (in this case you need the Artin motives as well) and $\bar{\mathbb F}_q$. Roughly, the outline is as follows.

If the Tate conjecture is true and the Frobenius action is semisimple, then the homological standard conjecture is true. Then Jannsen's Motives, numerical equivalence, and semi-simplicity implies that the category of motives with homological equivalence is semisimple. Moreover, the Tate conjecture gives a description of the simple objects in terms of the characteristic polynomial of Frobenius. Then Honda–Tate theory constructs all such characteristic polynomials inside the cohomology of an abelian variety. These are always dominated by Jacobians, so all desired Frobenius eigenvalues occur in the $H^1$ of a curve.

A great general reference for motives is the AMS Proceedings of Symposia in Pure Mathematics Vol. 55.1: Motives. See here for the AMS ebook collection version (you might be able to access this from within your institution). See in particular Proposition 2.6 of Milne's chapter Motives over finite fields.

Remark. As I understand it, the same is not expected to hold over basically any other type of field. For example, it should be easy to write down a Hodge structure that is not in the full abelian tensor subcategory generated by the Hodge structures of weight $1$.

This phenomenon is specific to the fields $\mathbb F_q$ (in this case you need the Artin motives as well) and $\bar{\mathbb F}_q$. Roughly, the outline is as follows.

If the Tate conjecture is true and the Frobenius action is semisimple, then the homological standard conjecture is true. Then Jannsen's Motives, numerical equivalence, and semi-simplicity implies that the category of motives with homological equivalence is semisimple. Moreover, the Tate conjecture gives a description of the simple objects in terms of the characteristic polynomial of Frobenius. Then Honda–Tate theory constructs all such characteristic polynomials inside the cohomology of an abelian variety. These are always dominated by Jacobians, so all desired Frobenius eigenvalues occur in the $H^1$ of a curve.

A great general reference for motives is the AMS Proceedings of Symposia in Pure Mathematics Vol. 55.1: Motives. See here for the AMS ebook collection version (you might be able to access this from within your institution). See in particular Proposition 2.6 and Remark 2.7 of Milne's chapter Motives over finite fields.

Remark. As I understand it, the same is not expected to hold over basically any other type of field. For example, it should be "easy" to write down a Hodge structure that is not in the full abelian tensor subcategory generated by the Hodge structures of weight $1$.

This phenomenon is specific to the fields $\mathbb F_q$ and $\bar{\mathbb F}_q$. Roughly, the outline is as follows.

If the Tate conjecture is true, then the homological standard conjecture is true, so the category of motives is semisimple by Jannsen's work. Moreover, the Tate conjecture gives a description of the simple objects in terms of the characteristic polynomial of Frobenius. Then Honda–Tate theory constructs all such characteristic polynomials inside the cohomology of an abelian variety. These are always dominated by Jacobians, so all desired Frobenius eigenvalues occur in the $H^1$ of a curve.

A great general reference for motives is the AMS Proceedings of Symposia in Pure Mathematics Vol. 55.1: Motives. See here for the AMS ebook collection version (you might be able to access this from within your institution). See in particular Proposition 2.6 of Milne's chapter Motives over finite fields.

Remark. As I understand it, the same is not expected to hold over basically any other type of field. For example, it should be easy to write down a Hodge structure that is not in the full abelian tensor subcategory generated by the Hodge structures of weight $1$.

This phenomenon is specific to the fields $\mathbb F_q$ (in this case you need the Artin motives as well) and $\bar{\mathbb F}_q$. Roughly, the outline is as follows.

If the Tate conjecture is true and the Frobenius action is semisimple, then the homological standard conjecture is true. Then Jannsen's Motives, numerical equivalence, and semi-simplicity implies that the category of motives with homological equivalence is semisimple. Moreover, the Tate conjecture gives a description of the simple objects in terms of the characteristic polynomial of Frobenius. Then Honda–Tate theory constructs all such characteristic polynomials inside the cohomology of an abelian variety. These are always dominated by Jacobians, so all desired Frobenius eigenvalues occur in the $H^1$ of a curve.

A great general reference for motives is the AMS Proceedings of Symposia in Pure Mathematics Vol. 55.1: Motives. See here for the AMS ebook collection version (you might be able to access this from within your institution). See in particular Proposition 2.6 of Milne's chapter Motives over finite fields.

Remark. As I understand it, the same is not expected to hold over basically any other type of field. For example, it should be easy to write down a Hodge structure that is not in the full abelian tensor subcategory generated by the Hodge structures of weight $1$.